A Spectral Framework for Closed-Form Relative Density Estimation

• URL: http://arxiv.org/abs/2605.10668v1
• Date: Mon, 11 May 2026 14:51:17 GMT
• Title: A Spectral Framework for Closed-Form Relative Density Estimation
• Abstract summary: We propose a closed-form spectral framework for relative log-density estimation in linearly parameterized probabilistic models.<n>We derive an explicit spectral formula based only on first- and second-order feature moments, yielding closed-form estimators of both divergences and log-density potentials for fixed features.
• Score: 0.554780083433538
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a closed-form spectral framework for relative log-density estimation in linearly parameterized probabilistic models, including unnormalized and conditional models. This is achieved by representing the Kullback-Leibler (KL) divergence as an integral of weighted chi-squared divergences, converting KL estimation into a family of least-squares problems. We derive an explicit spectral formula based only on first- and second-order feature moments, yielding closed-form estimators of both divergences and log-density potentials for fixed features. The framework extends to a broad class of f-divergences and can be combined with kernelization or feature learning with neural networks. We prove convergence guarantees for the resulting estimators and empirically compare them on synthetic data with optimization-based variational formulations, including logistic and softmax regression for normalized conditional models.

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